Abstract

In this paper we propose and analyze a stochastic collocation method for
solving the second order wave equation with a random wave speed and
subjected to deterministic boundary and initial conditions. The speed is
piecewise smooth in the physical space and depends on a finite number
of random variables. The numerical scheme consists of a finite
difference or finite element method in the physical space and a
collocation in the zeros of suitable tensor product orthogonal
polynomials (Gauss points) in the probability space. This approach leads
to the solution of uncoupled deterministic problems as in the Monte
Carlo method. We consider both full and sparse tensor product spaces of
orthogonal polynomials. We provide a rigorous convergence analysis and
demonstrate different types of convergence of the probability error with
respect to the number of collocation points for full and sparse tensor
product spaces and under some regularity assumptions on the data. In
particular, we show that, unlike in elliptic and parabolic problems, the
solution to hyperbolic problems is not in general analytic with respect
to the random variables. Therefore, the rate of convergence may only be
algebraic. An exponential/fast rate of convergence is still possible
for some quantities of interest and for the wave solution with
particular types of data. We present numerical examples, which confirm
the analysis and show that the collocation method is a valid alternative
to the more traditional Monte Carlo method for this class of problems.